Taylor expansion of metric

This is an exercise in Riemannian geometry. In this note I Taylor-expand the Riemannian metric in a normal neighborhood around a point in a Riemannian manifold M. What I have done is nothing except providing a few more terms than most standard textbooks. Hopefully the computation is correct. (I haven’t checked against other sources. )

Theorem 1 In a normal coordinates neighborhood of {p\in M}, the Taylor series of {g} around {p} is given by

\displaystyle  \begin{array}{rcl}  g_{ij}(x)&=& \delta_{ij} -\frac 1 3 R_{iklj}x^kx^l -\frac 1 6 R_{iklj;m} x^kx^lx^m\\ &&+ (\frac2{45} R_{ilmk}R_{jpqk}- \frac 1 {20} R_{ilmj;pq})x^lx^m x^p x^q\\ &&+(-\frac 1{90} R_{iklj;mpq}+\frac 2{45} R_{iklr;m}R_{jpqr})x^kx^lx^mx^px^q\\ && +(-\frac 1 {504}R_{iklj;mpqr}+ \frac{17}{1260}R_{ikls;pq}R_{jmps}+ \frac{11}{1008}R_{ikls;q}R_{jmps;r}\\ && +\frac 1{315}R_{ilms}R_{jqrt}R_{kspt})x^kx^lx^mx^px^qx^r +O(|x|^7). \end{array}

Proof: In normal coordinates, fix {x=(x^1, \cdots, x^n)} for the moment and let {\gamma = (t x^1, \cdots,t x^n)} be a radial geodesic, then {J(t)= t W^i \partial _i} is a Jacobi field on {\gamma(t)}, where {W^i\in \mathbb{R}}. (This can be seen by observing that tW is the variational vector field of the s-family of geodesics \gamma_s(t)= \exp_p (t(x+sW)), with a slight abuse of notations. ) Let {f= \langle J, J\rangle }.

We will use \cdot to denote a covariant derivative w.r.t. t and define {RX:= R(\dot\gamma, X)\dot \gamma}, then {\langle R(\dot\gamma, X)\dot\gamma,Y\rangle= \langle R(\dot \gamma, Y)\dot \gamma, X\rangle} implies {R} is a symmetric operator. Differentiating {\langle RX,Y\rangle=\langle RY,X\rangle} implies {\dot R, \ddot R,\cdots} are also symmetric. Also, the Jacobi equation simply becomes

\displaystyle \ddot J= RJ.

We compute (a way to test the correctness of the following computation is to note that each {R} carries two derivatives)

\displaystyle  \begin{array}{rcl}  f&=&\langle J,J\rangle.\\ f'&=& 2\langle \dot J,J\rangle.\\ f''&=& 2\langle RJ, J\rangle+2\langle \dot J,\dot J\rangle.\\ f'''&=& 2\langle \dot R J ,J\rangle +8\langle R\dot J,J \rangle.\\ f^{(4)}&=& 2\langle \ddot R J,J\rangle +(4+8)\langle \dot R \dot J, J\rangle +8 \langle RRJ, J\rangle + 8\langle R \dot J,\dot J\rangle\\ &=& 2\langle \ddot R J,J\rangle +12\langle \dot R \dot J, J\rangle +8 \langle RJ, RJ\rangle + 8\langle R \dot J, \dot J\rangle \end{array}

\displaystyle  \begin{array}{rcl}  f^{(5)} &=&2\langle \dddot R J,J\rangle +(4+12) \langle \ddot R \dot J, J\rangle +12 \langle \dot R RJ,J\rangle +(12+8) \langle \dot R\dot J,\dot J\rangle\\ &&+16 \langle \dot R J,RJ\rangle +16 \langle R\dot J, RJ\rangle +16 \langle RRJ,\dot J\rangle\\ &=&2\langle \dddot R J,J\rangle +16 \langle \ddot R \dot J, J\rangle +28 \langle \dot R J,RJ\rangle +20 \langle \dot R\dot J,\dot J\rangle +32\langle R\dot J, RJ\rangle \end{array}

\displaystyle  \begin{array}{rcl}  f^{(6)}&=&2 \langle \ddddot R J,J\rangle +(4+16) \langle \dddot R\dot J,J\rangle +16\langle \ddot R RJ,J\rangle +(16+20) \langle \ddot R \dot J,\dot J\rangle +28 \langle \ddot RJ,RJ\rangle\\ &&+(28+32) \langle \dot R \dot J, RJ\rangle +28\langle \dot R J, \dot R J\rangle + (28+32) \langle \dot R J, R\dot J\rangle +40 \langle \dot R RJ, \dot J\rangle\\ &&+32\langle RRJ,RJ\rangle +32\langle R\dot J, R\dot J\rangle\\ &=& 2 \langle \ddddot R J,J\rangle +20 \langle \dddot R\dot J,J\rangle +16\langle \ddot R RJ,J\rangle +36 \langle \ddot R \dot J,\dot J\rangle +28 \langle \ddot RJ,RJ\rangle+60 \langle \dot R \dot J, RJ\rangle\\ &&+28\langle \dot R J, \dot R J\rangle + 60 \langle \dot R J, R\dot J\rangle +40 \langle \dot R RJ, \dot J\rangle +32\langle RRJ,RJ\rangle +32\langle R\dot J, R\dot J\rangle\\ &=& 2 \langle \ddddot R J,J\rangle +20 \langle \dddot R\dot J,J\rangle +44\langle \ddot RJ,RJ\rangle +36 \langle \ddot R \dot J,\dot J\rangle+100 \langle \dot R \dot J, RJ\rangle +28\langle \dot R J, \dot R J\rangle\\ &&+60 \langle \dot R J, R\dot J\rangle +32\langle RRJ,RJ\rangle +32\langle R\dot J, R\dot J\rangle \end{array}

\displaystyle  \begin{array}{rcl}  f^{(7)}&=& 2 \langle R^{(5)} J,J\rangle + (4+20)\langle \ddddot R \dot J,J\rangle +20 \langle \dddot R RJ,J\rangle +(20+36) \langle \dddot R\dot J, \dot J\rangle + 44\langle \dddot RJ, RJ\rangle\\ && +( 44+100)\langle \ddot R\dot J, RJ\rangle + (44+56)\langle \ddot R J, \dot RJ\rangle +(44+60 )\langle \ddot R J, R \dot J\rangle +72 \langle \ddot R RJ, \dot J\rangle\\ && + (100+32)\langle \dot R RJ,RJ\rangle +(100+56+60) \langle \dot R \dot J , \dot R J\rangle +(100+60+64 )\langle \dot R \dot J, R \dot J\rangle\\ && +60 \langle \dot R J, RRJ\rangle +64 \langle R\dot R J, RJ\rangle +64 \langle RR\dot J, RJ\rangle + 64\langle RRJ, R\dot J\rangle\\ &=& 2 \langle R^{(5)} J,J\rangle + 24\langle \ddddot R \dot J,J\rangle +20 \langle \dddot R RJ,J\rangle +56 \langle \dddot R\dot J, \dot J\rangle + 44\langle \dddot RJ, RJ\rangle +144\langle \ddot R\dot J, RJ\rangle\\ && + 100\langle \ddot R J, \dot RJ\rangle +104\langle \ddot R J, R \dot J\rangle +72 \langle \ddot R RJ, \dot J\rangle + 132\langle \dot R RJ,RJ\rangle +(216 \langle \dot R \dot J , \dot R J\rangle\\ && +224\langle \dot R \dot J, R \dot J\rangle +60 \langle \dot R J, RRJ\rangle +64 \langle R\dot R J, RJ\rangle +64 \langle RR\dot J, RJ\rangle + 64\langle RRJ, R\dot J\rangle\\ &=& 2 \langle R^{(5)} J,J\rangle + 24\langle \ddddot R \dot J,J\rangle +20 \langle \dddot R RJ,J\rangle +56 \langle \dddot R\dot J, \dot J\rangle + 44\langle \dddot RJ, RJ\rangle +216\langle \ddot R\dot J, RJ\rangle\\ && + 100\langle \ddot R J, \dot RJ\rangle +104\langle \ddot R J, R \dot J\rangle + 132\langle \dot R RJ,RJ\rangle +(216 \langle \dot R \dot J , \dot R J\rangle +224\langle \dot R \dot J, R \dot J\rangle\\ && +60 \langle \dot R J, RRJ\rangle +64 \langle R\dot R J, RJ\rangle +128 \langle RR\dot J, RJ\rangle \end{array}

(Actually to compute {f^{(i)}(0)} we don’t have to compute all the terms of {f^{(i)}(t)}: we can omit the term containing any {J=J(0)=0}). It follows that

\displaystyle  \begin{array}{rcl}  f(0)&=&0.\\ f'(0)&=&0.\\ f''(0)&=&\langle W,W\rangle.\\ f'''(0)&=& 0.\\ f^{(4)}(0)&=& 8\langle W,RW\rangle=-8 \langle R(W,\dot \gamma)\dot \gamma, W\rangle.\\ f^{(5)}(0)&=&20\langle W, \dot R W\rangle=-20\langle \nabla_t R(\dot \gamma, W)W, \dot \gamma\rangle .\\ f^{(6)}(0)&=& 32\langle RW,RW\rangle+36\langle W, \ddot RW\rangle =32\langle R(W,\dot \gamma, )\dot\gamma, R(W,\dot \gamma)\dot \gamma\rangle-36 \langle \nabla ^2_t R (W, \dot \gamma)\dot \gamma , W\rangle.\\ f^{(7)}(0)&=&56 \langle \dddot R W, W\rangle + 224 \langle \dot R W, RW\rangle\\ &=&-56 \langle \nabla^3_t R (W,\dot \gamma)\dot \gamma,W\rangle+ 224 \langle \nabla_tR (W,\dot \gamma)\dot \gamma, R(W,\dot \gamma)\dot\gamma\rangle.\\ f^{(8)}(0) &=& (24+56)\langle \ddddot R W,W\rangle +(216+104+224)\langle \ddot RW,RW\rangle +(216+224)\langle \dot R W,\dot RW\rangle\\ && +128\langle RRW, RW\rangle\\ &=& 80\langle \ddddot R W,W\rangle +544\langle \ddot RW,RW\rangle +440\langle \dot R W,\dot RW\rangle +128\langle RRW, RW\rangle \end{array}

All the above expression are evaluated at {0}, noting that {J(0)=0}. So we have (all repeated indices are summed over):

\displaystyle  \begin{array}{rcl}  f(t)&=& t^2 g_{ij}(tx)W^i W^j\\ &=& \displaystyle\sum_{i=0}^7 \frac 1 {i!}f^{(i)}(0)t^i+O(t^8)\\ &=& \delta _{ij} t^2W^i W^j - \frac 13R_{iklj} x^kx^l t^4W^i W^j -\frac 1 6 R_{iklj;m}x^kx^lx^mt^5W^iW^j\\ &&+(\frac 2 {45}R_{ilmk}R_{jpqk}-\frac 1 {20} R_{ilmj;pq} )x^lx^mx^px^qt^6W^iW^j\\ && +(-\frac 1{90} R_{iklj;mpq}+\frac 2{45} R_{iklr;m}R_{jpqr})x^kx^lx^mx^px^qt^7W^iW^j\\ && +(-\frac 1 {504}R_{iklj;mpqr}+ \frac{17}{1260}R_{ikls;pq}R_{jmps}+ \frac{11}{1008}R_{ikls;q}R_{jmps;r}\\ && +\frac 1{315}R_{ilms}R_{jqrt}R_{kspt})x^kx^lx^mx^px^qx^rt^8W^iW^j +O(t^9). \end{array}

We thus conclude that

\displaystyle  \begin{array}{rcl}  g_{ij}(x)&=& \delta_{ij} -\frac 1 3 R_{iklj}x^kx^l -\frac 1 6 R_{iklj;m} x^kx^lx^m\\ &&+ (\frac2{45} R_{ilmk}R_{jpqk}- \frac 1 {20} R_{ilmj;pq})x^lx^m x^p x^q\\ &&+(-\frac 1{90} R_{iklj;mpq}+\frac 2{45} R_{iklr;m}R_{jpqr})x^kx^lx^mx^px^q\\ && +(-\frac 1 {504}R_{iklj;mpqr}+ \frac{17}{1260}R_{ikls;pq}R_{jmps}+ \frac{11}{1008}R_{ikls;q}R_{jmps;r}\\ &&+\frac 1{315}R_{ilms}R_{jqrt}R_{kspt})x^kx^lx^mx^px^qx^r +O(|x|^7). \end{array}

Here {|x|=r} is the radial distance from {p}. \Box

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5 Responses to Taylor expansion of metric

  1. Ken Leung says:

    The lengths of the formulae are remarkable. Thanks for KKK’s hard work in typesetting them.

  2. Salem says:

    in other places (for example http://arxiv.org/abs/1206.1092) the O(x^2) term has the opposite sign to the result here (j and l exchanged)

  3. KKK says:

    While I haven’t checked the higher order terms, the terms up to the forth order are correct, see e.g. Hamilton’s Ricci Flow p.59. The reason for the difference is that I have used a different convention for the Riemannian curvature tensor (my apology for not stating it): my definition is R_{ijkl} = \langle (\nabla _{e_i}\nabla _{e_j}- \nabla _{e_j}\nabla _{e_i}-\nabla _{[e_i,e_j]})e_k,e_l\rangle.

  4. Salem says:

    Thanks!

  5. zidane says:

    which can answer to my question: how to calculate the inverse of the metric g_ {ij}, in other words how to express g ^ {ij} -Thank you

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